Cyclotomic rational Cherednik algebras and categorifications (II)

This will be the second part of a joint talk with P. Shan. We will present a proof of Varagnolo–Vasserot's conjecture on the equivalence between the category $O$ of rational Cherednik algebras and a parabolic category $O$ of affine Lie algebras and some applications of this result (joint work with R. Rouquier and E. Vasserot).
In this talk, we will speak about categorifications and we explain another ingredient of the proof, that is reduction to codimension one. We will also present two corollaries of the result which involve the category $O$ of cyclotomic rational Cherednik algebra: this category is Koszul and the multiplicities of simples in standard modules are given by parabolic KL-polynomials (the first was conjectured by Chuang and Miyachi and the second by Rouquier).